Derivation

Split input into 2 regimes
if x < -0.02453297975411125 or 0.024108214702017676 < x
1. Initial program 0.1
  \[\frac{x - \sin x}{x - \tan x}\]
if -0.02453297975411125 < x < 0.024108214702017676
1. Initial program 62.8
  \[\frac{x - \sin x}{x - \tan x}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}\]
3. Applied simplify0.0
  \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*}\]
4. Using strategy rm
5. Applied fma-udef0.0
  \[\leadsto x \cdot \left(x \cdot \frac{9}{40}\right) - \color{blue}{\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
6. Applied associate--r+0.0
  \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \frac{9}{40}\right) - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}}\]
Recombined 2 regimes into one program.
Applied simplify0.0
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.02453297975411125 \lor \neg \left(x \le 0.024108214702017676\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \end{array}}\]

Runtime

herbie shell --seed 2018199 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

In
Out